A sequence is a function of the position n, and two tests name it: subtract neighbors for a common difference, divide them for a common ratio. Recursive rules and explicit rules for the same list, and how to turn one into the other. Series and sigma notation; Gauss’s pairing for 1 + 2 + … + 100, the finite geometric sum and the subtraction that proves it, infinite series that settle on a number and repeating decimals as fractions. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Four terms of a list: 10, 15, 20, 25. Each one sits a fixed distance from the last.
To reach the nth term you take n − 1 steps, not n. That one subtraction is what the formula a₁ + (n − 1)d is for — and forgetting it is the single most common mistake in the whole unit.
What people get wrong
⚠️People often think…
In 10, 15, 20, … the 8th term is 10 + 8(5) = 50.
The first term costs no steps. Getting to the 8th takes SEVEN: 10 + 7(5) = 45. Count the gaps, never the terms.
n − 1 steps to the nth term.
⚠️People often think…
In aₙ = 5n + 2 the first term is 2.
The 2 is where the rule would start at n = 0, and the list starts at n = 1. Plug it in: a₁ = 5(1) + 2 = 7. Always test the rule before you trust it.
Plug in n = 1. Do not read it off.
⚠️People often think…
The differences of 1, 4, 9, 16 follow a pattern, so it is arithmetic.
Arithmetic needs EQUAL differences, not interesting ones. These go 3, 5, 7 — and the ratios are not equal either, so it is neither arithmetic nor geometric. It is the squares.
Equal differences, or it is not arithmetic.
⚠️People often think…
a₁/(1 − r) gives the sum of any infinite geometric series.
Only when |r| < 1, so the terms shrink toward nothing.
Check |r| < 1 before you sum forever.
Subtract, divide, decide, check
1Watch one
Find the 8th term of 10, 15, 20, …
Subtract to find d: 15 − 10 = 5, and it holds.
Getting to the 8th term takes 8 − 1 = 7 steps.
a₈ = 10 + 7(5).
a₈ = 45. Count up if you doubt it: 10, 15, 20, 25, 30, 35, 40, 45. ✓
2Do one with me
Use aₙ = 5n + 2 and the series 1 + 2 + 4 + … + 512.
The first term a₁ is
Steps needed to reach a₈ is
The sum 1 + 2 + 4 + … + 512 is
💬One sentence, then you move on
Why is it n − 1 steps and not n?
3Try one
Find S₁₀ for 10, 15, 20, … where a₁₀ = 55. Use n(a₁ + aₙ)/2.
I want a hint first
10 + 55 = 65. Ten terms: 10 × 65 = 650. Now the step most people forget.
💬Last one — then you're done here
Why does an infinite sum only work when the terms shrink?
Where this goes
Where this lives
Saving the same amount every month is arithmetic; a balance earning the same percent is geometric. Which one you are in decides whether the total grows steadily or takes off.
What this feeds
Next unit is statistics, where a list of numbers becomes a claim about a whole group.
Name something that goes up by the same amount each time.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 From a list to a rule — the routine
Subtract neighbors. Same answer every time? Arithmetic, d = that answer, aₙ = a₁ + (n − 1)d. If not, divide neighbors. Same answer every time? Geometric, r = that answer, aₙ = a₁ · rⁿ⁻¹. Neither? Say “neither” and look for a different pattern (squares, Fibonacci). Then CHECK the rule on a term you already know: for 3, 7, 11, 15 the rule 4n − 1 gives a₃ = 11 ✓.
Subtract, divide, decide, check.
💡 The whole unit in one idea
A sequence is a function of the position n. Arithmetic means “add d each step” — a line, so the rule is linear in n. Geometric means “multiply by r each step” — exponential, so the rule has rⁿ⁻¹. A series just adds the terms up, and both sum formulas come from one trick each: pair the ends (arithmetic) or subtract rS from S (geometric).
⚠️ Traps the test loves
Using n instead of n − 1: a₈ of 10, 15, 20, … is 10 + 7(5) = 45, not 10 + 8(5) = 50.
Reading the constant in 5n + 2 as the first term. Plug in n = 1: a₁ = 7.
Forgetting the ÷ 2 in Sₙ = n(a₁ + aₙ)/2 — the answer doubles.
Giving the NEXT term as the sum: 1 + 2 + … + 512 is 1023, not 1024.
Using a₁/(1 − r) when |r| ≥ 1. The series diverges; the formula gives nonsense.
Calling a sequence arithmetic because the differences “follow a pattern.” They must be EQUAL: 1, 4, 9, 16 is neither.
📐 The formulas — the chart
Kind
Recursive
Explicit
Sum of n terms
Arithmetic (add d)
a₁ given; aₙ = aₙ₋₁ + d
aₙ = a₁ + (n − 1)d
Sₙ = n(a₁ + aₙ)/2
Geometric (multiply by r)
a₁ given; aₙ = r · aₙ₋₁
aₙ = a₁ · rⁿ⁻¹
Sₙ = a₁(1 − rⁿ)/(1 − r), r ≠ 1
Infinite geometric
—
needs |r| < 1
S = a₁/(1 − r)
Example: 3, 7, 11, 15
a₁ = 3, aₙ = aₙ₋₁ + 4
aₙ = 4n − 1; a₁₀ = 39
S₁₀ = 10(3 + 39)/2 = 210
Example: 2, 6, 18, 54
a₁ = 2, aₙ = 3 · aₙ₋₁
aₙ = 2 · 3ⁿ⁻¹; a₅ = 162
S₅ = 2(1 − 3⁵)/(1 − 3) = 242
🎯 How the test will ask
A list of four terms — “arithmetic, geometric or neither? Find d or r.” Subtract, then divide.
“Write a recursive rule and an explicit rule.” The recursive one must include a₁.
“Find a₂₅” or “which term is 99?” — the explicit rule, forward or solved for n.
A sigma expression — expand the first terms, count them, pick the sum formula.
A word problem with rows, weeks or bounces — decide first: a TERM (row 15) or a SUM (all 15 rows)?
An endless series — check |r| < 1 before you use a₁/(1 − r).
“Show why the formula works” — Gauss’s pairs, or S minus rS.
✅ Before the test, can you…
Tell arithmetic from geometric from neither in under a minute, with the test you used?
Write both rules for 3, 7, 11, 15 and turn one into the other?
Find a₅₀ without listing 49 terms?
Add 1 + 2 + … + 100 by pairing, and 1 + 2 + 4 + … + 512 by the subtraction trick?
Expand and evaluate a sigma expression?
Say when an infinite geometric series has a sum, and find it?
Turn 0.777… into 7/9 with a series?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: sort twelve sequences by the test that names them, turn a recursive rule into an explicit one step by step, graph a doubling sequence and read its rule off the bars, then argue that ten terms add to 1023 without adding them. Every activity checks itself, and hints are free.
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